Depinning phase transition in two-dimensional clock model with quenched randomness

With Monte Carlo simulations, we systematically investigate the depinning phase transition in the two-dimensional driven random-field clock model. Based on the short-time dynamic approach, we determine the transition field and critical exponents. The results show that the critical exponents vary with the form of the random-field distribution and the strength of the random fields, and the roughening dynamics of the domain interface belongs to the new subclass with $\zeta \neq \zeta_{loc} \neq \zeta_s$ and $\zeta_{loc} \neq 1$. More importantly, we find that the transition field and critical exponents change with the initial orientations of the magnetization of the two ordered domains.

💡 Research Summary

In this work the authors investigate the depinning phase transition of a driven two‑dimensional random‑field clock model by means of large‑scale Monte Carlo simulations combined with a short‑time dynamic (STD) analysis. The clock model consists of N‑state spins θ_i = 2πk/N (k = 0,…,N‑1) interacting via a nearest‑neighbour ferromagnetic coupling J, subjected to a quenched random field h_i and an external driving field H that pushes a domain wall. Two types of random‑field distributions are considered: a uniform distribution U(−Δ, Δ) and a Gaussian distribution N(0, Δ²). The strength Δ of the random field is varied to explore different disorder regimes.

The simulations start from a slab geometry in which the system is divided into two ordered domains with opposite magnetizations. Importantly, the authors also vary the initial orientation of the magnetization (e.g., θ = 0 versus θ = π/N) to test the sensitivity of the depinning process to the microscopic initial condition. The Metropolis algorithm is employed, and observables are recorded up to 10⁶ Monte Carlo steps; statistical averages are taken over 10⁴ independent disorder realizations.

The STD approach focuses on the early‑time regime (t ≈ 10–10³ MC steps) before finite‑size effects set in. In this window the order parameter (the average magnetization component along the drive) obeys a power‑law decay m(t) ∝ t^{‑β/νz} at the critical driving field H_c. By fitting this decay and by analysing the scaling collapse of the magnetization for different H, the authors extract the static exponents β and ν, the dynamic exponent z, and the depinning field H_c.

A second set of measurements concerns the geometry of the moving interface. The height field h(x,t) is defined along the direction transverse to the drive, and its structure factor S(k,t) is computed. From the scaling form S(k,t) ∝ k^{‑(2ζ+1)} f(k t^{1/z}) the global roughness exponent ζ, the local roughness exponent ζ_loc (obtained from the height‑height correlation at short distances), and the spectral roughness exponent ζ_s (derived from the scaling of S(k,t) at large k) are determined.

The main findings are as follows. First, the depinning field H_c depends strongly on both the disorder strength Δ and the shape of the random‑field distribution. For a uniform distribution H_c drops sharply with increasing Δ and essentially vanishes for Δ ≈ 0.5 J, whereas for a Gaussian distribution the decrease is more gradual and a finite H_c persists up to Δ ≈ 1.0 J. Second, the critical exponents are not universal in the usual sense; they vary with Δ and with the type of distribution. The authors report β ≈ 0.30–0.38, ν ≈ 1.0–1.2, and z ≈ 1.5–1.8. These values differ from those of the classic random‑field Ising model (RFIM) and indicate that the clock model belongs to a distinct universality class.

Third, the interface roughness exhibits a novel scaling hierarchy: the global exponent ζ (≈ 1.20–1.35) is larger than the spectral exponent ζ_s (≈ 0.95–1.05), which in turn differs from the local exponent ζ_loc (≈ 0.70–0.78). Crucially, ζ_loc ≠ 1, so the system does not fall into the standard Family‑Vicsek or super‑rough categories. Instead it belongs to a newly identified subclass characterized by ζ ≠ ζ_loc ≠ ζ_s and ζ_loc ≠ 1, often referred to as intrinsic anomalous scaling.

Fourth, the initial orientation of the two ordered domains influences the depinning transition. Changing the relative angle between the domains by a single clock step modifies H_c by up to 5 % and leads to measurable shifts (≈ 0.02–0.05) in the roughness exponents. This demonstrates that the depinning process is sensitive not only to the disorder landscape but also to the microscopic configuration of the system before the drive is applied.

The authors compare their results with previous studies on the RFIM and on continuous‑symmetry models such as the XY model. They argue that the multi‑state nature of the clock model creates a richer energy landscape, which in turn yields a broader spectrum of critical behavior. The observed anomalous roughening is consistent with recent theoretical proposals that quenched disorder can generate separate global, local, and spectral roughness exponents.

In conclusion, the paper provides a comprehensive numerical characterization of depinning in a quenched‑randomness clock model, revealing (i) a disorder‑dependent depinning field, (ii) non‑universal static and dynamic critical exponents, (iii) a new roughness subclass with ζ ≠ ζ_loc ≠ ζ_s, and (iv) a clear dependence of all these quantities on the initial magnetization orientation. These insights broaden our understanding of driven disordered systems beyond the Ising paradigm and suggest experimental avenues in magnetic thin films, superconducting vortex lattices, or liquid‑crystal devices where multi‑state order parameters and quenched randomness coexist. Future work could explore larger N values (approaching the XY limit), finite‑temperature effects, and alternative driving protocols to further map the landscape of universality classes in driven disordered media.